I was reading https://en.wikipedia.org/wiki/Exponential_sum
In this article under “estimates” they say that the exponential sum is $O(\sqrt N)$ and thus resembles a random walk. Is there any reason to think of this sum as a random walk other than just a $\sqrt N$?
Best Answer
I think the wikipedia section means this:
To the extent that the $x_n$ sequence can be considered random, each of $y_n = e(x_n) = \exp(2\pi i x_n)$ is a unit vector with a random direction, so the successive sums $y_1, y_1+y_2, y_1+y_2+y_3, \dots$ is a random walk on the $2$-D plane where each step is size $1$ with random direction.